Transcript Properties of the MIMO radar ambiguity function

Properties of the MIMO Radar
Ambiguity Function
Chun-Yang Chen and P. P. Vaidyanathan
California Institute of Technology
Electrical Engineering/DSP Lab
ICASSP 2008
Outline
 Review of the background
– Radar ambiguity function and its properties
– MIMO radar
– MIMO radar ambiguity function
 Properties of the MIMO ambiguity function
–
–
–
–
Signal component
Energy
Symmetry
Linear frequency modulation (LFM)
 Conclusion
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
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Review: Ambiguity function and MIMO radar
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Radar Ambiguity Function
u(t)
u(t-t)ej2pnt
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
t: delay
n: Doppler
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Radar Ambiguity Function
u(t)
Matched filter
output

u(t-t)ej2pnt
t: delay
n: Doppler
(u(t t )e j 2pnt )(u(t t ' )e j 2pn 't )* dt
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
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Radar Ambiguity Function
u(t-t)ej2pnt
u(t)
Matched filter
output

t: delay
n: Doppler
(u(t t )e j 2pnt )(u(t t ' )e j 2pn 't )* dt
  u(t )u (t  (t t ' ))e
*
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
j 2p (n n ') t
dt
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Radar Ambiguity Function
u(t-t)ej2pnt
u(t)
Matched filter
output

(u(t t )e j 2pnt )(u(t t ' )e j 2pn 't )* dt
  u(t )u (t  (t t ' ))e
*
Radar ambiguity
function
t: delay
n: Doppler
j 2p (n n ') t
dt
 (t ,n )   u(t )u* (t  t )e j 2pnt dt
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
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Radar Ambiguity Function
u(t-t)ej2pnt
u(t)
Matched filter
output

(u(t t )e j 2pnt )(u(t t ' )e j 2pn 't )* dt
  u(t )u (t  (t t ' ))e
*
Radar ambiguity
function
t: delay
n: Doppler
j 2p (n n ') t
dt
 (t ,n )   u(t )u* (t  t )e j 2pnt dt
 Ambiguity function characterizes the Doppler and range
resolution.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
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Radar Ambiguity Function
u(t )
Multiple targets
(tk,nk)
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
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Radar Ambiguity Function
u(t )
K
j 2pn k t
u
(
t

t
)
e

k
k 1
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
Multiple targets
(tk,nk)
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Radar Ambiguity Function
u(t )
K
j 2pn k t
u
(
t

t
)
e

k
k 1
Matched filter
output
K

k 1
k
Multiple targets
(tk,nk)
  (t  t k ,n n k )
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
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Radar Ambiguity Function
u(t )
K
j 2pn k t
u
(
t

t
)
e

k
k 1
Matched filter
output
K

k 1
k
Multiple targets
(tk,nk)
  (t  t k ,n n k )
n
target 1 (t1,n1)
target 2 (t2,n2)
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
t
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Radar Ambiguity Function
u(t )
K
j 2pn k t
u
(
t

t
)
e

k
k 1
Matched filter
output
n
K

k 1
k
Multiple targets
(tk,nk)
  (t  t k ,n n k )
 (t t1 ,n n 1 )
target 1 (t1,n1)
target 2 (t2,n2)
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
t
13
Radar Ambiguity Function
 Ambiguity function characterizes the Doppler and range
resolution.
n
 (t t1 ,n n 1 )
target 1 (t1,n1)
target 2 (t2,n2)
t
 (t ,n )   u (t )u (t  t ) e

j 2pn t
dt
Ambiguity function
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
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Radar Ambiguity Function
 Ambiguity function characterizes the Doppler and range
resolution.
n
 (t t1 ,n n 1 )
target 1 (t1,n1)
target 2 (t2,n2)
t
 (t ,n )   u (t )u (t  t ) e

j 2pn t
dt
Ambiguity function
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
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Properties of Radar Ambiguity Function
 Signal component
 (0,0)  1   (t ,n )
n
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
t
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Properties of Radar Ambiguity Function
 Signal component
 (0,0)  1   (t ,n )
 Energy

2
 (t ,n ) dtdn  1
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
n
t
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Properties of Radar Ambiguity Function
 Signal component
 (0,0)  1   (t ,n )
 Energy

2
 (t ,n ) dtdn  1
 Symmetry
n
t
 (t ,n )   (t ,n )
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
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Properties of Radar Ambiguity Function
 Signal component
 (0,0)  1   (t ,n )
 Energy

2
 (t ,n ) dtdn  1
n
 Symmetry
t
 (t ,n )   (t ,n )
 Linear frequency modulation (LFM)
u
LFM
(t )  u(t )e
jpkt 2
 LFM (t ,n )   (t ,n  kt )
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
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MIMO Radar
The radar systems which emits orthogonal (or noncoherent)
waveforms in each transmitting antennas are called MIMO radar.
MIMO radar
f2(t)
f1(t)
f0(t)
SIMO radar (Traditional)
w2f(t)
w1f(t)
w0f(t)
 Advantages
– Better spatial resolution [Bliss & Forsythe 03]
– Flexible transmit beampattern design [Fuhrmann & San Antonio 04]
– Improved parameter identifiability [Li et al. 07]
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
Ambiguity Function in MIMO Radar
(t,n,f) t:delay
n:Doppler
f: Spatial freq.
TX
dT
u0(t) u1(t)
…
uM-1(t)
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
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Ambiguity Function in MIMO Radar
(t,n,f)
TX
dT
u0(t) u1(t)
t:delay
n:Doppler
f: Spatial freq.
RX
…
uM-1(t)
(t,n,f)
…
dR
MF
…
MF
…
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
MF
…
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Ambiguity Function in MIMO Radar
(t,n,f)
TX
dT
u0(t) u1(t)
t:delay
n:Doppler
f: Spatial freq.
RX
…
uM-1(t)
(t,n,f)
…
dR
MF
…
MF
…
y
(t ,n , f )
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
MF
…
(t )
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Ambiguity Function in MIMO Radar
(t,n,f)
TX
RX
…
dT
u0(t) u1(t)
t:delay
n:Doppler
f: Spatial freq.
uM-1(t)
Matched filter output
 (y
(t ',n ', f ')
(t )
)
H
(t,n,f)
…
dR
MF
…
MF
…
y
(t ,n , f )
MF
…
(t )
 y (t ,n , f ) (t )dt
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
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Ambiguity Function in MIMO Radar
Matched filter output
 (y

(t ',n ', f ')
(t )
)
H
y
(t ,n , f )
N 1
M 1 M 1
n 0
m  0 m '0
(t )dt
t:delay
n:Doppler
f: Spatial freq.
um(t): m-th waveform
xm: m-th antenna location
n: receiving antenna index
(
)
j 2p ( f  f ') n
*
j 2p (n v ') t
j 2p ( fm f 'm ')
e

u
(
t

t
)
u
(
t

t
'
)
e
dt
e

  m
m
Receiver beamforming
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
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Ambiguity Function in MIMO Radar
Matched filter output
 (y

(t ',n ', f ')
(t )
)
H
y
(t ,n , f )
N 1
M 1 M 1
n 0
m  0 m '0
(t )dt
t:delay
n:Doppler
f: Spatial freq.
um(t): m-th waveform
xm: m-th antenna location
n: receiving antenna index
(
)
j 2p ( f  f ') n
*
j 2p (n v ') t
j 2p ( fm f 'm ')
e

u
(
t

t
)
u
(
t

t
'
)
e
dt
e

  m
m
Receiver beamforming
 m ,m ' (t ,n )   um (t )um* ' (t  t )e j 2pn t dt
Cross ambiguity function
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
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Ambiguity Function in MIMO Radar
Matched filter output
 (y

(t ',n ', f ')
(t )
)
H
y
(t ,n , f )
N 1
M 1 M 1
n 0
m  0 m '0
(t )dt
t:delay
n:Doppler
f: Spatial freq.
um(t): m-th waveform
xm: m-th antenna location
n: receiving antenna index
(
)
j 2p ( f  f ') n
*
j 2p (n v ') t
j 2p ( fm f 'm ')
e

u
(
t

t
)
u
(
t

t
'
)
e
dt
e

  m
m
Receiver beamforming
 m ,m ' (t ,n )   um (t )um* ' (t  t )e j 2pn t dt
[San Antonio et al. 07]
M 1 M 1
 (t ,n , f , f ' )     m,m' (t ,n )e j 2p ( fm f 'm')
m  0 m ' 0
MIMO ambiguity function
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
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Properties of the MIMO ambiguity function
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Properties of the signal component
 Ambiguity function:
 Signal component:
 (t ,n , f , f ' )
 (0,0, f , f )
 (0,0, f , f ' )
f ' f
f'
 (0,0, f , f )
f
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
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Properties of the signal component
 Ambiguity function:
 Signal component:
 (0,0, f , f ' )
 (t ,n , f , f ' )
 (0,0, f , f )
For orthogonal waveforms,

um (t )um* ' (t )dt   mm '
f ' f
f'
 (0,0, f , f )
f
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
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Properties of the signal component
 Ambiguity function:
 Signal component:
 (0,0, f , f ' )
 (t ,n , f , f ' )
 (0,0, f , f )
For orthogonal waveforms,

f ' f
f'
 (0,0, f , f )
um (t )um* ' (t )dt   mm '
  (0,0, f , f )  M , f
If the waveforms are orthogonal,
the signal component will be a
constant for all angle.
f
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
31
Properties of the signal component
 Ambiguity function:
 Signal component:
 (t ,n , f , f ' )
 (0,0, f , f )
For general waveforms,

For orthogonal waveforms,
um (t ) dt  1
2

um (t )um* ' (t )dt   mm '
  (0,0, f , f )  M , f
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
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Properties of the signal component
 Ambiguity function:
 Signal component:
 (t ,n , f , f ' )
 (0,0, f , f )
For general waveforms,

For orthogonal waveforms,
um (t ) dt  1
2
dT
 If
is integer,


um (t )um* ' (t )dt   mm '
  (0,0, f , f )  M , f
   (0,0, f , f ) df  M , f
The integration of the signal
component is a constant if dT is
a multiple of the wavelength.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
dT is the spacing
between the
transmitting antennas
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Properties of the signal component
 Ambiguity function:
 Signal component:
 (t ,n , f , f ' )
 (0,0, f , f )
For general waveforms,

dT is the spacing
between the
transmitting antennas
For orthogonal waveforms,

um (t ) dt  1
2
dT
 If
is integer,

um (t )um* ' (t )dt   mm '
  (0,0, f , f )  M , f
   (0,0, f , f ) df  M , f
 For the general case,
2dT /   M   (0,0, f , f ) df  2dT /   M

2d T / 
2d T / 
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
In general, the
integration of the
signal component is
confined.
34
Energy of the cross ambiguity function
 Cross ambiguity function:
 mm ' (t ,n )   um (t )um* ' (t  t )e j 2pnt dt
 Energy of the cross ambiguity function:

 mm ' (t ,n ) dt dn
2
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
35
Energy of the cross ambiguity function
 Cross ambiguity function:
 mm ' (t ,n )   um (t )um* ' (t  t )e j 2pnt dt
 Energy of the cross ambiguity function:

 
 mm ' (t ,n ) dt dn
2
u
m
(t )u (t  t )e
*
m'
j 2pn t
2
dt dndt
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
36
Energy of the cross ambiguity function
 Cross ambiguity function:
 mm ' (t ,n )   um (t )um* ' (t  t )e j 2pnt dt
 Energy of the cross ambiguity function:

 
 mm ' (t ,n ) dt dn
2
u
m
(t )u (t  t )e
*
m'
j 2pn t
2
dt dndt
   um (t )u (t  t ) dtdt
*
m'

2
Parserval
relation
( u (t) dt)  1
2
2
m
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
37
Energy of the cross ambiguity function
 Cross ambiguity function:
 mm ' (t ,n )   um (t )um* ' (t  t )e j 2pnt dt
 Energy of the cross ambiguity function:

 mm ' (t ,n ) dt dn  1
2
The energy of the cross
ambiguity function is a
constant.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
38
Energy of the MIMO ambiguity function
 MIMO ambiguity function:
M 1 M 1
 (t ,n , f , f ' )     mm ' (t ,n )e j 4pd
T
/  ( fm f 'm ')
m  0 m ' 0
 Energy of the ambiguity function
     (t ,n , f , f ' )
2
dtdndfdf '
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
39
Energy of the MIMO ambiguity function
 MIMO ambiguity function:
M 1 M 1
 (t ,n , f , f ' )     mm ' (t ,n )e j 4pd
T
/  ( fm f 'm ')
m  0 m ' 0
 Energy of the ambiguity function
     (t ,n , f , f ' )
 
2
dtdndfdf '
2
M 1 M 1
j 4pd

(
t
,
n
)
e
  mm '
T
/  ( fm f 'm ')
dfdf ' dtdn
m  0 m ' 0
dT is the spacing
between the
transmitting antennas
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
40
Energy of the MIMO ambiguity function
 MIMO ambiguity function:
M 1 M 1
 (t ,n , f , f ' )     mm ' (t ,n )e j 4pd
T
/  ( fm f 'm ')
m  0 m ' 0
 Energy of the ambiguity function
     (t ,n , f , f ' )
 
2
dtdndfdf '
2
M 1 M 1
j 4pd

(
t
,
n
)
e
  mm '
m  0 m ' 0
M 1 M 1
dT is the spacing
between the
transmitting antennas
T
/  ( fm f 'm ')
      mm ' (t ,n ) dtdn
2
m  0 m ' 0
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
dfdf ' dtdn
If dT is a multiple of
the wavelength, we
can apply Parserval
relation for 2D DFT.
41
Energy of the MIMO ambiguity function
 MIMO ambiguity function:
M 1 M 1
 (t ,n , f , f ' )     mm ' (t ,n )e j 4pd
T
/  ( fm f 'm ')
m  0 m ' 0
 Energy of the ambiguity function
     (t ,n , f , f ' )
 
2
dtdndfdf '
2
M 1 M 1
j 4pd

(
t
,
n
)
e
  mm '
m  0 m ' 0
M 1 M 1
T
/  ( fm f 'm ')
dfdf ' dtdn
      mm ' (t ,n ) dtdn
2
m  0 m ' 0
M 1 M 1
  1  M
m  0 m ' 0
2
Cross ambiguity
function has
constant energy
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
dT is the spacing
between the
transmitting antennas
42
Energy of the MIMO ambiguity function
 If dT is a multiple of the wavelength,

 (t ,n , f , f ' ) dtdndfdf '  M 2
2
dT is the spacing
between the
transmitting antennas
If dT is a multiple of the
wavelength, the energy of
the MIMO ambiguity function
is a constant.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
43
Energy of the MIMO ambiguity function
 If dT is a multiple of the wavelength,

 (t ,n , f , f ' ) dtdndfdf '  M 2
2
dT is the spacing
between the
transmitting antennas
 Recall that the signal component satisfies,
  (0,0, f , f ) df
 M , f
– Because energy and the signal component are both constants,
we can only spread the energy to minimize the peak.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
44
Energy of the MIMO ambiguity function
 If dT is a multiple of the wavelength,

 (t ,n , f , f ' ) dtdndfdf '  M 2
2
dT is the spacing
between the
transmitting antennas
 In general, the energy satisfies,
2d T /  
2
2dT /   M 2 

2

(
t
,
n
,
f
,
f
'
)
d
t
d
n
dfdf
'

M

(2dT /  )2
(2dT /  )2
2
2
In general, the energy of the
MIMO ambiguity function is
confined in a certain range.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
45
Energy of the MIMO ambiguity function
 If dT is a multiple of the wavelength,

 (t ,n , f , f ' ) dtdndfdf '  M 2
2
dT is the spacing
between the
transmitting antennas
 In general, the energy satisfies,
2d T /  
2
2dT /   M 2 

2

(
t
,
n
,
f
,
f
'
)
d
t
d
n
dfdf
'

M

(2dT /  )2
(2dT /  )2
2
2
 In general, the signal component satisfies,
2dT /   M   (0,0, f , f ) df  2dT /   M

2d T / 
2d T / 
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
46
Symmetry properties
 Symmetry of the cross ambiguity function
 mm ' (t ,n )   m'm (t ,n )
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
47
Symmetry properties
 Symmetry of the cross ambiguity function
 mm ' (t ,n )   m'm (t ,n )
 Symmetry of the MIMO ambiguity function
 (t ,n , f , f ' )   (t ,n , f ' , f )
It suffices to show only half of
the ambiguity function (t>0).
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
48
Linear frequency modulation (LFM)
 Linear frequency modulation
u
LFM
m
(t )  um (t )e
jpkt 2
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
49
Linear frequency modulation (LFM)
 Linear frequency modulation
u
LFM
m
(t )  um (t )e
jpkt 2
 Cross ambiguity function
LFM
 mm
' (t ,n )   mm ' (t ,n  kt )
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
50
Linear frequency modulation (LFM)
 Linear frequency modulation
u
LFM
m
(t )  um (t )e
jpkt 2
 Cross ambiguity function
LFM
 mm
' (t ,n )   mm ' (t ,n  kt )
 MIMO ambiguity function

LFM
Shear off
(t ,n , f , f ' )   (t ,n  kt , f , f ' )
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
51
Linear frequency modulation (LFM)
 (t ,n , f , f ' )
n
t
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
52
Linear frequency modulation (LFM)
 (t ,n  kt , f , f ' )
 (t ,n , f , f ' )
LFM
n
Shear off
t
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
n
t
53
Linear frequency modulation (LFM)
 (t ,n  kt , f , f ' )
 (t ,n , f , f ' )
LFM
Shear off
n
The range
resolution is
improved by
LFM.
t
n
n
t
n
t
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
t
54
Conclusion
 Properties of the MIMO ambiguity function
– Signal component
2dT /   M   (0,0, f , f ) df  2dT /   M

2d T / 
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
2d T / 
55
Conclusion
 Properties of the MIMO ambiguity function
– Signal component
2dT /   M   (0,0, f , f ) df  2dT /   M

2d T / 
– Energy
2d T / 
2
2dT /   M 2 
2dT /   M 2

(
t
,
n
,
f
,
f
'
)
d
t
d
n
dfdf
'


(2dT /  )2
(2dT /  )2
2
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
2
56
Conclusion
 Properties of the MIMO ambiguity function
– Signal component
2dT /   M   (0,0, f , f ) df  2dT /   M

2d T / 
– Energy
– Symmetry
2d T / 
2
2dT /   M 2 
2dT /   M 2

(
t
,
n
,
f
,
f
'
)
d
t
d
n
dfdf
'


(2dT /  )2
(2dT /  )2
2
2
 (t ,n , f , f ' )   (t ,n , f ' , f )
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
57
Conclusion
 Properties of the MIMO ambiguity function
– Signal component
2dT /   M   (0,0, f , f ) df  2dT /   M

2d T / 
– Energy
– Symmetry
– LFM
2d T / 
2
2dT /   M 2 
2dT /   M 2

(
t
,
n
,
f
,
f
'
)
d
t
d
n
dfdf
'


(2dT /  )2
(2dT /  )2
2
2
 (t ,n , f , f ' )   (t ,n , f ' , f )
 LFM (t ,n , f , f ' )   (t ,n  kt , f , f ' )
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
58
Thank You!
Q&A
Any questions?
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
59
Properties of the signal component
If the waveforms are
orthogonal, the
signal component
will be a constant
for all angle.
For orthogonal waveforms,
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

um (t )um* ' (t )dt   mm '
  (0,0, f , f )  M , f
60
Properties of the signal component
For general waveforms,

um (t ) dt  1
2
dT
 If
is integer,

   (0,0, f , f ) df  M , f
The integration of
the signal
component is a
constant if dT is a
multiple of the
wavelength.
dT is the spacing
between the
transmitting antennas
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
61
Properties of the signal component
In general, the
integration of the signal
component is confined
in a certain range.
 For the general case,
2dT /   M   (0,0, f , f ) df  2dT /   M

2d T / 
2d T / 
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
dT is the spacing
between the
transmitting antennas
62
MIMO Radar
TX
RX
…
SIMO
Radar
…
MF
MF
MF
u (t)
RX
TX
MIMO
Radar
…
u0(t) u1(t)
…
uM-1(t)
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
MF
…
…
MF
…
…
MF
…
…
63
MIMO Radar
 Advantages
– Better spatial resolution [Bliss & Forsythe 03]
– Flexible transmit beampattern design [Fuhrmann & San Antonio 04]
– Improved parameter identifiability [Li et al. 07]
RX
TX
MIMO
Radar
…
u0(t) u1(t)
…
uM-1(t)
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
MF
…
…
MF
…
…
MF
…
…
64